Setting the pivot point of an object
You need to either change the origin of the object, or use the 3D Cursor as a pivot point.

A quick way to change what is used as a pivot point, is to look at the bottom menu in the 3d view and select "Pivot center for rotation/scaling":



But to actually change the object's origin, you could first move the 3D Cursor to whereever you want the pivot point. To do this accurately, you can snap it to whatever you have selected (both in Object Mode and Edit Mode by pressing Shift-S and choosing "Cursor to selected".




Then, in Object Mode with you object selected, use the Object Tools menu on the left (T to make the menu appear if you don't see it), press Origin button, and choose "Origin to 3D Cursor".

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For Blender 2.8:

The 'Pivot Point' selector can be found at the top of your viewport in a drop-down with a double 'chain-link' icon on it